# Chapter Nine by mifei

VIEWS: 8 PAGES: 13

• pg 1
```									                CHAPTER 9: BOND PRICES AND YIELDS

1.   a.   Effective annual rate on three-month T-bill:
4
 100,000 
           1  (1.02412) 4  1  0.1000  10.00%
 97,645 

b.   Effective annual interest rate on coupon bond paying 5% semiannually:
(1.05)2 – 1 = 0.1025 = 10.25%
Therefore, the coupon bond has the higher effective annual interest rate.

2.   The effective annual yield on the semiannual coupon bonds is 8.16%. If the annual
coupon bonds are to sell at par they must offer the same yield, which requires an
annual coupon of 8.16%.

3.   The bond callable at 105 should sell at a lower price because the call provision is
more valuable to the firm. Therefore, its yield to maturity should be higher.

4.   The bond price will be lower. As time passes, the bond price, which is now above
par value, will approach par.

5.   True. Under the expectations hypothesis, there are no risk premia built into bond
prices. The only reason for long-term yields to exceed short-term yields is an
expectation of higher short-term rates in the future.

6.   c.   A “fallen angel” is a bond that has fallen from investment grade to junk
bond status.

7.   Uncertain. Lower inflation usually leads to lower nominal interest rates.
Nevertheless, if the liquidity premium is sufficiently great, long-term yields can
exceed short-term yields despite expectations of falling short rates.

8.   If the yield curve is upward sloping, you cannot conclude that investors expect
short-term interest rates to rise because the rising slope could be due to either
expectations of future increases in rates or the demand of investors for a risk
premium on long-term bonds. In fact the yield curve can be upward sloping even in
the absence of expectations of future increases in rates.

9-1
9.    a.    The bond pays \$50 every six months.
Current price:
[\$50  Annuity factor(4%, 6)] + [\$1000  PV factor(4%, 6)] = \$1,052.42
Assuming the market interest rate remains 4% per half year, price six months
from now:
[\$50  Annuity factor(4%, 5)] + [\$1000  PV factor(4%, 5)] = \$1,044.52

b.    Rate of return =
\$50  (\$1,044 .52  \$1,052 .42 ) \$50  \$7.90
             0.0400  4.00 % per six months
\$1,052 .42              \$1,052 .42

10. a.      Use the following inputs: n = 40, FV = 1000, PV = –950, PMT = 40. You
will find that the yield to maturity on a semi-annual basis is 4.26%. This
implies a bond equivalent yield to maturity of: 4.26%  2 = 8.52%
Effective annual yield to maturity = (1.0426)2 – 1 = 0.0870 = 8.70%

b.    Since the bond is selling at par, the yield to maturity on a semi-annual basis is
the same as the semi-annual coupon, 4%. The bond equivalent yield to
maturity is 8%.
Effective annual yield to maturity = (1.04)2 – 1 = 0.0816 = 8.16%

c.    Keeping other inputs unchanged but setting PV = –1050, we find a bond
equivalent yield to maturity of 7.52%, or 3.76% on a semi-annual basis.
Effective annual yield to maturity = (1.0376)2 – 1 = 0.0766 = 7.66%

11.   Since the bond payments are now made annually instead of semi-annually, the
bond equivalent yield to maturity is the same as the effective annual yield to
maturity. The inputs are: n = 20, FV = 1000, PV = –price, PMT = 80. The
resulting yields for the three bonds are:
Bond equivalent yield =
Bond Price
Effective annual yield
\$ 950                8.53%
\$1000                8.00%
\$1050                7.51%
The yields computed in this case are lower than the yields calculated with semi-annual
coupon payments. All else equal, bonds with annual payments are less attractive to
investors because more time elapses before payments are received. If the bond price is
the same with annual payments, then the bond's yield to maturity is lower.

9-2
12.
Inflation in                       Coupon        Principal
Time                            Par value
year just ended                      payment      repayment
0                            \$1,000.00
1              2%            \$1,020.00         \$40.80          0
2              3%            \$1,050.60         \$42.02          0
3              1%            \$1,061.11         \$42.44     \$1,061.11
Interest  Pr ice appreciation
Nominal return =
Initial price
1  Nominal return
Real return =                       1
1  Inflation

Second year                              Third year
\$42.02  \$30.60                   \$42.44  \$10.51
Nominal return:                             0.071196                         0.050400
\$1020                           \$1050.60
1.071196                            1.05040
Real return:                1  0.0400  4.00%               1  0.0400  4.00%
1.03                               1.01

The real rate of return in each year is precisely the 4% real yield on the bond.

13.   Remember that the convention is to use semi-annual periods:
Maturity       Maturity         Semi-annual   Bond equivalent
Price
(years)       (half-years)         YTM             YTM
\$400.00       20.00           40.00            2.317%          4.634%
\$500.00       20.00           40.00            1.748%          3.496%
\$500.00       10.00           20.00            3.526%          7.052%
\$376.89       10.00           20.00            5.000%         10.000%
\$456.39       10.00           20.00            4.000%          8.000%
\$400.00       11.68           23.36            4.000%          8.000%

14.
Zero 8% coupon 10% coupon
a. Current prices                  \$463.19 \$1000     \$1134.20
b. Price one year from now         \$500.25 \$1000     \$1124.94
Price increase                  \$ 37.06 \$ 0.00     -\$ 9.26
Coupon income                   \$ 0.00 \$80.00      \$ 100.00
Income                          \$ 37.06 \$80.00     \$ 90.74
Rate of Return                   8.00% 8.00%        8.00%

9-3
15.   The reported bond price is: 100 2/32 percent of par = \$1,000.625
However, 15 days have passed since the last semiannual coupon was paid, so
accrued interest equals: \$35  (15/182) = \$2.885
The invoice price is the reported price plus accrued interest: \$1003.51

16. If the yield to maturity is greater than current yield, then the bond offers the prospect
of price appreciation as it approaches its maturity date. Therefore, the bond is selling
below par value.

17. The coupon rate is below 9%. If coupon divided by price equals 9%, and price is less
than par, then coupon divided by par is less than 9%.

18. The solution is obtained using Excel:
A                          B          C          D                E
1                                         5.50% coupon bond,
2                                         maturing March 15, 2014
3                                                     Formula in Column B
4   Settlement date                         2/22/2006 DATE(2006,2,22)
5   Maturity date                           3/15/2014 DATE(2014,3,15)
6   Annual coupon rate                          0.055
7   Yield to maturity                          0.0534
8   Redemption value (% of face value)            100
9   Coupon payments per year                        2
10
11
12   Flat price (% of par)                      101.03327   PRICE(B4,B5,B6,B7,B8,B9)
13   Days since last coupon                           160   COUPDAYBS(B4,B5,2,1)
14   Days in coupon period                            181   COUPDAYS(B4,B5,2,1)
15   Accrued interest                             2.43094   (B13/B14)*B6*100/2
16   Invoice price                              103.46393   B12+B15

9-4
19. The solution is obtained using Excel:
A           B              C      D          E           F         G
1                                                 Semiannual             Annual
2                                                 coupons                coupons
3
4     Settlement date                             2/22/2006              2/22/2006
5     Maturity date                               3/15/2014              3/15/2014
6     Annual coupon rate                              0.055                  0.055
7     Bond price                                        102                    102
8     Redemption value (% of face value)                100                    100
9     Coupon payments per year                            2                      1
10
11    Yield to maturity (decimal)                  0.051927               0.051889
12
13
14    Formula in cell E11:              YIELD(E4,E5,E6,E7,E8,E9)

20. a.      The maturity of each bond is 10 years, and we assume that coupons are paid
semiannually. Since both bonds are selling at par value, the current yield to
maturity for each bond is equal to its coupon rate.
If the yield declines by 1%, to 5% (2.5% semiannual yield), the Sentinal bond
will increase in value to 107.79 [n=20; i = 2.5%; FV = 100; PMT = 3]
The price of the Colina bond will increase, but only to the call price of 102.
The present value of scheduled payments is greater than 102, but the call price
puts a ceiling on the actual bond price.

b.     If rates are expected to fall, the Sentinal bond is more attractive: since it is not
subject to being called, its potential capital gains are higher.
If rates are expected to rise, Colina is a better investment. Its higher coupon
(which presumably is compensation to investors for the call feature of the
bond) will provide a higher rate of return than the Sentinal bond.

c.     An increase in the volatility of rates increases the value of the firm’s option to
call back the Colina bond. [If rates go down, the firm can call the bond, which
puts a cap on possible capital gains. So, higher volatility makes the option to
call back the bond more valuable to the issuer.] This makes the Colina bond
less attractive to the investor.

9-5
21. The price schedule is as follows:
Remaining Constant yield value                      Imputed interest
Year
Maturity (T)  1000/(1.08)T                   (Increase in constant yield value)
0 (now) 20 years      \$214.55
1          19          231.71                                 \$17.16
2          18          250.25                                  18.54
19          1          925.93
20          0         1000.00                                  74.07

22. The bond is issued at a price of \$800. Therefore, its yield to maturity is 6.8245%.
Using the constant yield method, we can compute that its price in one year (when
maturity falls to 9 years) will be (at an unchanged yield) \$814.60, representing an
increase of \$14.60. Total taxable income is: \$40 + \$14.60 = \$54.60

23. a.      Initial price, P0 = 705.46 [n = 20; PMT = 50; FV = 1000; i = 8]
Next year's price, P1 = 793.29 [n = 19; PMT = 50; FV = 1000; i = 7]
\$50  (\$793.29  \$705.46)
HPR                               0.1954  19.54%
\$705.46

b.     Using OID tax rules, the cost basis and imputed interest under the constant yield
method are obtained by discounting bond payments at the original 8% yield to
maturity, and simply reducing maturity by one year at a time:

Constant yield prices: compare these to actual prices to compute capital gains
P0 = \$705.46
P1 = \$711.89 so implicit interest over first year = \$6.43
P2 = \$718.84 so implicit interest over second year = \$6.95
Tax on explicit plus implicit interest in first year
= 0.40  (\$50 + \$6.43) = \$22.57
Capital gain in first year = Actual price at 7% YTM – constant yield price
= \$793.29 – \$711.89 = \$81.40
Tax on capital gain = 0.30  \$81.40 = \$24.42
Total taxes = \$22.57 + \$24.42 = \$46.99

\$50  (\$793.29  \$705.46)  \$46.99
c.     After tax HPR                                        0.1288  12.88%
\$705.46

9-6
d.   Value of bond after two years equals \$798.82 [using n = 18; i = 7]
Total income from the two coupons, including reinvestment income:
(\$50  1.03) + \$50 = \$101.50
Total funds after two years: \$798.82 + \$101.50 = \$900.32
Therefore, the \$705.46 investment grows to \$900.32 after two years.
705.46  (1 + r)2 = 900.32  r = 0.1297 = 12.97%

e.   Coupon received in first year:                    \$50.00
Tax on coupon @ 40%                              – 20.00
Tax on imputed interest (0.40  \$6.43)           – 2.57
Net cash flow in first year                       \$27.43
If you invest the year-1 cash flow at an after-tax rate of:
3%  (1 – 0.40) = 1.8%
then, by year 2, it will grow to:
\$27.43  1.018 = \$27.92
You sell the bond in the second year for:   \$798.82
Tax on imputed interest in second year:      – 2.78           [0.40  \$6.95]
Coupon received in second year, net of tax: + 30.00           [\$50  (1 – 0.40)]
Capital gains tax on sales price:            – 23.99          [0.30  (\$798.82 – \$718.84)]
using constant yield value
CF from first year's coupon (reinvested):   + 27.92           [from above]
TOTAL                                  \$829.97
Thus, after two years, the initial investment of \$705.46 grows to \$829.97:
705.46  (1 + r)2 = 829.97  r = 0.0847 = 8.47%

24. a.   The bond sells for \$1,124.72 based on the 3.5% yield to maturity:
[n = 60; i = 3.5; FV = 1000; PMT = 40]
Therefore, yield to call is 3.368% semiannually, 6.736% annually:
[n = 10; PV = 1124.72; FV = 1100; PMT = 40]

b.   If the call price were \$1050, we would set FV = 1050 and redo part (a) to find
that yield to call is 2.976% semi-annually, 5.952% annually. With a lower call
price, the yield to call is lower.

c.   Yield to call is 3.031% semiannually, 6.062% annually:
[n = 4; PV = 1124.72 ; FV = 1100; PMT = 40]

9-7
25. The stated yield to maturity equals 16.075%:
[n = 10; PV = 900; FV = 1000; PMT = 140]
Based on expected coupon payments of \$70 annually,
the expected yield to maturity is: 8.526%

26.   The bond is selling at par value. Its yield to maturity equals the coupon rate, 10%. If
the first-year coupon is reinvested at an interest rate of r percent, then total proceeds at
the end of the second year will be: [100  (1 + r) + 1100]. Therefore, realized
compound yield to maturity will be a function of r as given in the following table:
r      Total proceeds     Realized YTM =       Pr oceeds / 1000  1
8%          \$1208         1208 / 1000  1  0.0991  9.91 %
10%          \$1210         1210 / 1000  1  0.1000  10 .00 %
12%          \$1212         1212 / 1000  1  0.1009  10 .09 %

27.   Zero coupon bonds provide no coupons to be reinvested. Therefore, the final value of the
investor's proceeds from the bond is independent of the rate at which coupons could be
reinvested (if they were paid). There is no reinvestment rate uncertainty with zeros.

28.   April 15 is midway through the semi-annual coupon period. Therefore, the invoice
price will be higher than the stated ask price by an amount equal to one-half of the
semiannual coupon. The ask price is 101.125 percent of par, so the invoice price is:
\$1,011.25 + (1/2  \$50) = \$1,036.25

29. Factors that might make the ABC debt more attractive to investors, therefore
justifying a lower coupon rate and yield to maturity, are:
i.    The ABC debt is a larger issue and therefore may sell with greater liquidity.
ii. An option to extend the term from 10 years to 20 years is favorable if interest rates
ten years from now are lower than today’s interest rates. In contrast, if interest
rates are rising, the investor can present the bond for payment and reinvest the
money for better returns.
iii. In the event of trouble, the ABC debt is a more senior claim. It has more
underlying security in the form of a first claim against real property.
iv. The call feature on the XYZ bonds makes the ABC bonds relatively more
attractive since ABC bonds cannot be called from the investor.
v.    The XYZ bond has a sinking fund requiring XYZ to retire part of the issue each
year. Since most sinking funds give the firm the option to retire this amount at
the lower of par or market value, the sinking fund can work to the detriment of
bondholders.

9-8
30. a.   The floating-rate note pays a coupon that adjusts to market levels. Therefore,
it will not experience dramatic price changes as market yields fluctuate. The
fixed rate note therefore will have a greater price range.

b.   Floating rate notes may not sell at par for any of the several reasons:
The yield spread between one-year Treasury bills and other money market
instruments of comparable maturity could be wider than it was when the bond
was issued.
The credit standing of the firm may have eroded relative to Treasury securities
that have no credit risk. Therefore, the 2% premium would become
insufficient to sustain the issue at par.
The coupon increases are implemented with a lag, i.e., once every year.
During a period of rising interest rates, even this brief lag will be reflected in
the price of the security.

c.   The risk of call is low. Because the bond will almost surely not sell for much
above par value (given its adjustable coupon rate), it is unlikely that the bond
will ever be called.

d.   The fixed-rate note currently sells at only 88% of the call price, so that yield to
maturity is above the coupon rate. Call risk is currently low, since yields
would have to fall substantially for the firm to use its option to call the bond.

e.   The 9% coupon notes currently have a remaining maturity of fifteen years and
sell at a yield to maturity of 9.9%. This is the coupon rate that would be
needed for a newly issued fifteen-year maturity bond to sell at par.

f.   Because the floating rate note pays a variable stream of interest payments to
maturity, its yield-to-maturity is not a well-defined concept. The cash flows
one might want to use to calculate yield to maturity are not yet known. The
effective maturity for comparing interest rate risk of floating rate debt
securities with other debt securities is better thought of as the next coupon
reset date rather than the final maturity date. Therefore, “yield-to-recoupon
date” is a more meaningful measure of return.

9-9
31. a.   (1)   Current yield = Coupon/Price = 70/960 = 0.0729 = 7.29%
(2)   YTM = 3.993% semiannually or 7.986% annual bond equivalent yield
[n = 10; PV = (-)960; FV = 1000; PMT = 35]
Then compute the interest rate.
(3) Realized compound yield is 4.166% (semiannually), or 8.332% annual
bond equivalent yield. To obtain this value, first calculate the future value of
reinvested coupons. There will be six payments of \$35 each, reinvested
semiannually at a per period rate of 3%:
[PV = 0; PMT = \$35; n = 6; i = 3%] Compute FV = \$226.39
The bond will be selling at par value of \$1,000 in three years, since coupon is
forecast to equal yield to maturity. Therefore, total proceeds in three years
will be \$1,226.39. To find realized compound yield on a semiannual basis
(i.e., for six half-year periods), we solve:
\$960  (1 + yrealized)6 = \$1,226.39  yrealized = 4.166% (semiannual)

b.   Shortcomings of each measure:
(1) Current yield does not account for capital gains or losses on bonds
bought at prices other than par value. It also does not account for reinvestment
income on coupon payments.
(2) Yield to maturity assumes that the bond is held to maturity and that all
coupon income can be reinvested at a rate equal to the yield to maturity.
(3) Realized compound yield (horizon yield) is affected by the forecast of
reinvestment rates, holding period, and yield of the bond at the end of the investor's
holding period.

32. a.   The yield to maturity of the par bond equals its coupon rate, 8.75%. All else
equal, the 4% coupon bond would be more attractive because its coupon rate
is far below current market yields, and its price is far below the call price.
Therefore, if yields fall, capital gains on the bond will not be limited by the
call price. In contrast, the 8.75% coupon bond can increase in value to at most
\$1050, offering a maximum possible gain of only 5%. The disadvantage of
the 8.75% coupon bond in terms of vulnerability to a call shows up in its
higher promised yield to maturity.

b.   If an investor expects rates to fall substantially, the 4% bond offers a greater
expected return.

c.   Implicit call protection is offered in the sense that any likely fall in yields
would not be nearly enough to make the firm consider calling the bond. In
this sense, the call feature is almost irrelevant.

9-10
33. Market conversion value = value if converted into stock = 20.83 \$28 = \$583.24
Conversion premium = Bond value – market conversion value
= \$775 – \$583.24 = \$191.76

34. a.    The call provision requires the firm to offer a higher coupon (or higher
promised yield to maturity) on the bond in order to compensate the investor
for the firm's option to call back the bond at a specified call price if interest
rates fall sufficiently. Investors are willing to grant this valuable option to the
issuer, but only for a price that reflects the possibility that the bond will be
called. That price is the higher promised yield at which they are willing to buy
the bond.

b.   The call option reduces the expected life of the bond. If interest rates fall
substantially so that the likelihood of call increases, investors will treat the bond
as if it will "mature" and be paid off at the call date, not at the stated maturity
date. On the other hand if rates rise, the bond must be paid off at the maturity
date, not later. This asymmetry means that the expected life of the bond will be
less than the stated maturity.

c.   The advantage of a callable bond is the higher coupon (and higher promised
yield to maturity) when the bond is issued. If the bond is never called, then an
investor will earn a higher realized compound yield on a callable bond issued at
par than on a non-callable bond issued at par on the same date. The
disadvantage of the callable bond is the risk of call. If rates fall and the bond is
called, then the investor receives the call price and will have to reinvest the
proceeds at interest rates that are lower than the yield to maturity at which the
bond was originally issued. In this event, the firm's savings in interest payments
is the investor's loss.

35. a.    The forward rate (f2) is the rate that makes the return from rolling over one-
year bonds the same as the return from investing in the two-year maturity bond
and holding to maturity:
1.08  (1 + f2) = (1.09)2  f2 = 0.1001 = 10.01%

b.   According to the expectations hypothesis, the forward rate equals the expected
value of the short-term interest rate next year, so the best guess would be 10.01%.

c.   According to the liquidity preference hypothesis, the forward rate exceeds the
expected short-term interest rate next year, so the best guess would be less
than 10.01%.

9-11
36.   The top row must be the spot rates. The spot rates are (geometric) averages of the
forward rates, and the top row is the average of the bottom row. For example, the
spot rate on a two-year investment (12%) is the average of the two forward rates
10% and 14.0364%:
(1.12)2 = 1.10  1.140364 = 1.2544

37. a.     We obtain forward rates from the following table:
Maturity
YTM                Forward rate         Price (for part c)
(years)
1        10.0%                                    \$909.09 (\$1000/1.10)
2        11.0% 12.01% [(1.112/1.10) – 1]          \$811.62 (\$1000/1.112)
3        12.0% 14.03% [(1.123/1.112) – 1]         \$711.78 (\$1000/1.123)

b.   We obtain next year’s prices and yields by discounting each zero’s face value
at the forward rates derived in part (a):
Maturity
Price                                       YTM
(years)
1         \$892.78     [ = 1000/1.1201]                 12.01%
2         \$782.93     [ = 1000/(1.1201 x 1.1403)]      13.02%
Note that this year’s upward sloping yield curve implies, according to the
expectations hypothesis, a shift upward in next year’s curve.

c.   Next year, the two-year zero will be a one-year zero, and it will therefore sell
at: \$1000/1.1201 = \$892.78
Similarly, the current three-year zero will be a two-year zero, and it will sell
for: \$782.93
Expected total rate of return:
\$892.78
two-year bond:            1  0.1000  10.00%
\$811.62
\$782.93
three-year bond:            1  0.1000  10.00%
\$711.78

9-12
38. a.   A three-year zero with face value \$100 will sell today at a yield of 6% and a
price of: \$100/1.063 =\$83.96
Next year, the bond will have a two-year maturity, and therefore a yield of 6%
(reading from next year’s forecasted yield curve). The price will be \$89.00,
resulting in a holding period return of 6%.

b.   The forward rates based on today’s yield curve are as follows:
Year                Forward Rate
2        [(1.052/1.04) – 1] = 6.01%
3        [(1.063/1.052)– 1] = 8.03%
Using the forward rates, the yield curve next year is forecast as:
Year                         Forward Rate
1                                       6.01%
2          [(1.0601  1.0803)1/2 – 1] = 8.03%

The market forecast is for a higher yield to maturity for two–year bonds than
your forecast. Thus, the market predicts a lower price and higher rate of return.

39. a.   (2) The dividends from the preferred stock are less secure than the interest
from the bond.

b.   (3) The yield on the callable bond must compensate the investor for the risk
of call.
Choice (1) is wrong because, although the owner of a callable bond receives
principal plus a premium in the event of a call, the interest rate at which he
can subsequently reinvest will be low. The low interest rate that makes it
profitable for the issuer to call the bond makes it a bad deal for the bond’s
holder.
Choice (2) is wrong because a bond is more apt to be called when interest
rates are low. There will be an interest saving for the issuer only if rates are
low.

c.   (3)
d.   (2)
e.   (4)

9-13

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